Stochastic Optimisation: An Introduction

It is a branch of Operational Research, which deals with the optimisation problems, where some or all of the model parameters are random. In real life, most of the constraint or outcome of a process is a random number. For example, when a company has to make a decision to make an investment in a subset of the projects, then it is neither sure about profit nor about the resources, which will be generated/require to complete each of the projects. Likewise, when a trader makes a decision some stock, then he is unaware of the precise gain from each of the stocks, which he can buy.

I am going to introduce three essential techniques to solve the Stochastic Optimisation Problems:

1] Mean-Variance Optimisation

2] Expected Value Method

3] Deterministic Approximation Approach

Mean-Variance Optimization: In this approach, we maximise the expected benefit (simple average of the returns) generated by the projects. At the same time, we reduce the risk (variance of the profits) involving in those projects. It is a challenging task. Many a time, financial managers fix the expected benefit at a fixed number (given by their customers) by adding an equality constraint. Furthermore, they minimise the risk.

Expected value method: In this method, we replace all the random numbers, involved in the problem by their expected (simple average/ mean) value. Next, we solve the transformed deterministic optimisation problem by using any appropriate algorithm of Operational Research. It is effortless and straightforward to apply in any real-life stochastic programming problem.

Deterministic Equivalent Approach: In this method, we consider many scenarios, which may occur in real life with some recourse action (the penalty levied to make infeasible solution feasible). For example, we estimated that we shall require 15 people to complete some project. However, after we started working on the project, we felt that we need 2 more people. We can hire two people, but we shall have to pay some money (penalty) to hire them. To solve the problem by underlying method, Firstly, we consider some/all of future outcomes (scenarios). Secondly, we assume that the final result is belonging to one of the scenarios under consideration. Thirdly, to transform the stochastic optimisation problem to deterministic equivalent, we introduce one constraint for each scenario. Also, we adjust the optimality function by considering the recourse action. Finally, we solve the deterministic optimisation problem by using any well known Operational Research algorithm.